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Problem Set #8, Solutions Econ 2106H, J.L. Turner 1. A monopolist produces where MR = MC in order to maximize profit. 2. MC D MR = region of CS when there is a monopolist. The entire shaded region is consumer surplus in the presence of competition. 3 a) Monopolist produces at MR = MC so 24 - 2Q = 2Q →6 and P = 24 - 6 →P = 18 The AC at Q = 6 is... AC = (Q2+4)/Q = Q + 4/Q = 6 + 4/6 = 6 2/3. Profit = (18-6 2/3)(6) = 68 3 Consumer Surplus = ½ (24-18)(6) = 18 b) In the perfectly competitive market, P = MC → 24 - Q = 2Q → Q = 8 when Q = 8, P = 24 - 8 = 16 Consumer Surplus = ½(24-16)(8) = 32 AC at Q =8 is... AC = Q2/Q + 4/Q = Q + 4/Q = 8 + 4/8 =8½ Profit = (16 - 8 ½)(8) = 60 c) AC for part a) = 6 2/3 b)= 8 ½ ( these were found above) d) First establish the competitive market equilibrium with 6 firms. For each firm MC = 2Qi , which we can solve for Qi = MC/2 The industry supply QM is Q1 + Q2+...+Q6 , which is QM = 3MC Which directly gives us QM = 3P, or P = (1/3)QM. This last equation is the supply equation. Industry equilibrium output sets Industry Supply = Industry Demand doing this yields... (1/3)QM = 24 - QM which can be solved for... QM = 18 Putting this into the demand equation P = 24-(18) = 16 Q*M = 18 P* = 6 (cont’d) i) Each firm sets MC = P in a competitive industry. So, 2Qi = 6 Qi = 3 ii) Market output is (3)(6) =18 iii) Equilibrium Price is 6. iv) Consumer Surplus = (24-6)(18)(1/2) = 162 v) Firm Profit = (P-Aci)(Qi) AC = Q2/Q + 4/Q = Q + 4/Q = 3 + 4/3 = 4 1/3 so, Profit = (6 - 4 1/3)(3) = 5 vi) Total Profits = 6*5 = 30 e) AC is below equilibrium price f) because of e) firms will continue to enter the market. g) this is done in exactly the same manner as previously. 4) Monopoly 1 Firm Competition 6 firm Competition Firm Profits 68 60 5 Consumer Surplus 18 32 162 Welfare 86 92 6*5+162 =192 5) a) AC = C/Q = (2+2Q2)/Q = 2Q + 2/Q b-d) The long run equilibrium quantity will be set where MC = AC. 4Q = 2/Q + 2Q 2Q = 2/Q Q2 = 1 Q=1 ( Q= -1 is ruled out on economic grounds...it doesn’t make sense) so each firm will produce one unit. In equilibrium (long run), profits are zero. By definition, the equation Profit = (P - AC)Qi = 0 represents the profits to a single firm. Using the demand curve, and the average cost curve, and denoting the market quantity as QM, we get... Profitsi = [(20-QM) - AC]Qi = 0 where Qi = the quantity produced by firm i. Note that the AC evaluated at the optimal firm output of 1 is 2(1) + 2/1 = 4. So, Profitsi = [(20-QM) - 4](1) = 0 which can be solved for... QM = 16 Since market output is 16, and each firm produces 1 unit, there are 16 firms in the industry. Since QM =16, and the demand curve is P = 20 -QM, the equilibrium price is 4. e) MC =4Qi To find the industry supply curve... MC = 4Qi Qi = (¼)MC, so Q1 + Q2 +...+Q20 = QM = (1/4)(20)MC =5MC QM= 5MC → QM = 5P → P = (1/5)QM which is the industry supply curve To find equilibrium, supply = demand: 20 - QM = (1/5)QM 20 = (1 1/5)QM → Q*M = 16 2/3 which, when put back into the demand equation, yields... P = 20 - (16 2/3) = 3 1/3 Each firm will produce where MC = P , which is expressed 4Qi = 3 1/3 Qi = 10/12 ≈ .833 At Qi = .833, AC = 2(.833) +2/.833 = 4.067 Profit = (P - AC)Qi = (3 1/3 - 4.067).833 = -.611 f) the same process is done as above. 6. a. $ 100 80 80 100 Q b. Students $32, Executives $50. c. $48. This is the are under the upper demand curve between Q=0 and Q=80. d. Solve 50 - P = 48 - 32, to get P = $34. If P is higher, Executives buy the student package. e. Students will pay at most $30. Executives would get $12 in net consumer surplus. Hence, the Journal can charge at most $38 for the 100-article package if it wants executives to buy it. f. It earns more with a 60-article student package. On average, it gets .5 (30) + .5 (38) = $34 if it does this. If it instead sold an 80-article student package, it would sell that for $32 and the 100-article package for $34, so the average price is $33 < $34. 7. a. A declining average cost of production. b. One firm can produce at lower average cost. c. Price CS (triangle above P* beneath demand curve) P* Demand MR Profit AC MC Quantity d. Price Demand MR P* Loss AC MC Quantity e. Price Demand P* MR AC MC Quantity